
\(Z(N|t)\)  (REF ) is a \(\tau \) -function of the Toda chain (as a formal series) with \(N\)  playing role of the discrete time [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, while, in the DV case, it is a sum of \(Z_a\)  (introduced in the previous subsection) which is this \(\tau \) -function [7]}.

\(Z_{\rm DV}(T_k,N_i)\)  determines the SW system; hence, it satisfies the Whitham hierarchy [8]}, [9]} in the planar limit, and
\(T_k\)  are Whitham flows [10]}, [11]}, [12]}.

In the planar limit, \(Z_{\rm DV}(T_k,N_i)\)  as a function of \(T_k\)  and \(N_i\)  also satisfies [10]}, [11]} the WDVV equations [15]}, [16]}, [17]}, which is typical for \(\tau \) -functions of Whitham hierarchies, and for the SW systems [18]}, [19]}, [20]}, [21]}.

The Dijkgraaf–Vafa partition function \(Z_{\rm DV}(T_k,N_i)\)  as the SW system is also associated with a many-body integrable system, classical [22]}, [23]}, [24]}, [25]}, [26]} in the planar limit or quantum [27]}, [28]}, [29]}, [30]}, maybe even with the Whitham flows quantized [27]} (see table in the Introduction). This quantization is realized by the check-operators.

